This is an excellent question.

For a good overview of connectedness in topology, see Section 2.4 in

http://sites.millersville.edu/rumble/StudentProjects/Schreffler/schreffler.pdf

For a thoroughgoing look at connectedness as well as disconnected in topology, see

http://www.sciencedirect.com/science/article/pii/0016660X75900094

For first step You can use

https://en.wikipedia.org/wiki/Connectedness

Thanks James for the very useful link to Arhangelskii and Wiegandt 's paper.

@Aleksandar Jovanovic: if the world is just digital 1 order of magnification decides the reality.

This is a very interesting observation.   There is a bit more one can observe, if we consider a topology on the space of descriptions of connected objects.   The basis for this topology would be a fundamental set of descriptions so that the description of every connected object is the union or the intersection of basis descriptions.   In this case, a description is a vector in R^n and the components of each vector are real numbers,  where each component is a feature value of a connected object.         

I would like to add that homology is a kind of measure of connectedness, or a measure of holes of the space. I understand that you are interested for general topology, in which case the references of Prof. Peters is what you need for a start.

Dear Friend

The notion of connectedness and other classical topological notions have been extensively studied in the framework of generalized topologies, minimal structures, via topological ideals and other structures. See authors as Akos Cszazar, Takashi Noiri, Carlos Carpintero, Ennis Rosas (you can find more information about these authors and his works in the following link http://www.ams.org/mrlookup )